Geomagnetic Storm Conditions

DENNIS ODIJK1 School of Geomatic Engineering, University of New South Wales, Sytlney NSW 2052 Australia

Due to the maximum of the solar cycle, ionospheric activ-

ity increased considerably last year. At frequent times

warnings were sent out announcing geomagnetic storms

disturbing the ionospheric electron content. In this article

the influence of such geomagnetic storms on fast and pre-

cise GPS positioning (for surveying applications at mid-

latitude regions) is studied. And here with "fast" it is aimed

at the shortest observation time possible: carrier ambigui-

ty resolution and position estimation using only one single

epoch of data. To apply this instantaneous data processing

technique successfully to GPS baselines of medium length

(up to 50 km), additional ionospheric information is

inevitable, not only under geomagnetic storm, but also

under more quiet conditions. However, in this article it will

be shown that under geomagnetic storm conditions, even

for rather short baselines (<10 km), for which the ionos-

pheric delays under more quiet conditions could be

neglected, one has to account for significant relative ionos-

pheric delays. Therefore an important facet of this contri-

bution is the investigation of how to process baselines of

varying length in a more uniform way, making use of a

permanent GPS network (if available in the surveying

area) and a stochastic modeling technique of the ionos-

pheric delays. © 2001 John Wiley & Sons, Inc.

INTRODUCTION!n 15 July 2000 a severe geomagnetic storm

occurred. This storm was one of the 30 strongest

disturbances since 1932 and degraded GPS single-point

positioning by about a factor of 3 during the storm days,

1On leave from Delft University of Technology, The Netherlands.

GPS Solutions, Vol. 5, No. 2, pp. 29-42 (2001)© 2001 John Wiley & Sons, Inc.

compared to the accuracy during days prior to the storm

(Kunches, 2000). In this article the manner in which high-

precision relative GPS positioning is affected by such

geomagnetic storms is discussed. This investigation is

restricted to geomagnetic mid-latitude regions where the

effect of geomagnetic storms is less compared to equato-

rial or auroral regions. In those regions geomagnetic

storms may cause ionospheric scintillation resulting in a

degraded GPS receiver performance (Skone & de Jong,

2000).

Attention is focused on instantaneous applications,

for which carrier phase ambiguity resolution and coordi-

nate estimation have to be performed using just one

epoch of GPS data. Instantaneous processing has a num-

ber of advantages over batch-type processing. Among

these there are the following:

9 the shortest observation time span possible is used

(important to real-time applications);

* no initialization process is needed to resolve the

ambiguities;9 the method is insensitive to cycle-slips or loss-of-

locks;9 the data processing is very suitable for moving

receivers (kinematic applications).

The instantaneous processing technique has already

been proved to be successful for short baselines (see, e.g.,

de Jong, Bock, & Bevis, 2000), neglecting the relative

ionospheric delays. Nowadays it is desirable that this

technique also suits baselines for which the ionospheric

delays may not be neglected.

Relative ionospheric delays are known to be the

largest errors for baselines longer than 10 km; for shorter

baselines they are normally so small that they may be

neglected in the processing. In this article it will be inves-

Instantaneous Precise OPS Positioning 28

tigated whether this is still the case under geomagneticstorm conditions. Test computations with data of the 15July storm day are compared to 14 July, one day prior to the

storm. For both days two GPS baselines are considered: ashort baseline of 4 km and one medium-length baseline of54 km. For both baselines the performance of instanta-

neous ambiguity resolution and position estimation willbe analyzed. All GPS data are taken from the Southern Cal-ifornia Integrated GPS Network (SCIGN), located in a geo-

magnetic mid-latitude region (<p = 33°N, A, = 243°E).

GEOMAGNETIC DISTURBANCESThe behavior of the Earth's magnetic field is a significantfactor for GPS measurements at mid-latitude regions.

The Planetary K-Index (Kp) is a measure of the geomag-netic activity. It is computed every 3 h and published, as

well as other parameters, on the Internet site(www.sec.noaa.gov) of the Space Environment Center

(SEC) of the U.S. National Oceanic and AtmosphericAdministration (NOAA).

According to the NOAA Space Weather Scale for Geo-magnetic Storms a maximum Kp-value of 9 coincideswith an extreme geomagnetic storm, occurring 4 times

per solar cycle on average. Figure 1 shows that on 15 July2000 from 15:00-24:00 UTC an extreme geomagneticstorm occurred. In this article how this 15 July geomag-netic storm affected precise GPS positioning will be dis-cussed. For the sake of comparison, data measured on

the day before (14 July) has also been included in theinvestigation. For this day, the geomagnetic disturbanceswere known to be less severe (see Figure 1), though from

15:00-21:00 there were also storm activities, although at arelatively minor level (Kp = 5-6). However, minor geo-magnetic storm disturbances are much more frequent

than severe ones: they happen on average 1700 times persolar cycle.

Short-Baseline Performance with the IonosphericDelays NeglectedIn relative GPS processing it is common practice toneglect errors in the data due to propagation through theionosphere when the baseline length is less than about10-20 km. In this section the results of an investigation as

to whether this practice is valid under the geomagneticconditions of 14 and 15 luly are presented. For this pur-pose data of two permanent stations within the SCIGNnetwork have been used: BGIS and DYHS, which are at a

distance of about 4 km from each other. For the purposeof this article the data of these permanent stations have

been processed as if they were a common GPS baselinemeasured by a user, and the coordinates of station DYHShave been assumed to be unknown. Five hours of dual-

frequency carrier phase and code (pseudo-range) datawere used, obtained from 19:00-24:00 UTC on 14 and 15July. This time interval corresponds to 11:00-16:00 localtime, when it is expected that the ionosphere has its diur-

nal maximum. Both stations are equipped with AshtechZ-XII3 receivers and collected the data with 30-s samplinginterval. The cut-off elevation was set at 10 degrees and

the number of tracked satellites was at every epoch largeror equal to 6; see Figure 2 where the number of satellitesis plotted against the time span for 14 July. Note that on 15

July this graph is the same, due to the daily repeating GPSconstellation, except for a shift of about 4 min.

The data processing (and all other computations

described in this article) was carried out using theGPSveQ software, courtesy of the Delft University of

Technology (see de Jonge, 1998).

Ambiguity ^solutionFor this baseline, the first step was to estimate the correctinteger ambiguities according to the normal "short base-

line model" in which the double-differenced (DD) ionos-pheric delays are assumed zero.

The instantaneous ambiguity resolution was carriedout with the LAMBDA method on an epoch-by-epoch

basis, using dual-frequency phase as well as code data.Hence, for each individual observation epoch (600 intotal), a new ambiguity solution was obtained, using only

the data of that epoch without taking information fromprevious epochs into account.

The mathematical adjustment model for this instan-

taneous ambiguity resolution, consisting of a functionaland stochastic model part, can, in a very simplified form,be written as follows, assuming that m satellites are

tracked by the two baseline receivers:

In model (1) the symbols used have the following

meaning (i= 1, 2):<j>r. "observed-minus-computed" DD phase observa-

tions on Li;

pi', "observed-minus-computed" DD code observa-

tions on Li;B: design matrix elements with satellite-receiver

geometry;

b: increments of baseline coordinates;A.J: wavelength of Li;

Im-i; identity matrix of dimension m-1;

at: DD phase ambiguities on Li;

dpi. variance of undifferenced phase observations on Li;

dp2: variance of undifferenced code observations on Li;

D: double-difference operator;®: Kronecker-product;

Note that the observations are corrected for tropos-

pheric delays using the Saastamoinen model.The reason of the factor 2 appearing in front of the

variance-covariance matrix of the observations in model

(1) is that the observations in the functional part are alldouble-differenced, whereas the variances are denoted

as undifferenced quantities. For all computationsdescribed in this article the undifferenced standard devi-

ations of the phase observations on LI as well as on L2

Instantaneous Precise GPS Positioning 31

have been assumed equally, namely O0t = (% = 3 mm. The

precision of the code observations is assumed to be a fac-tor 100 worse than the precision the phase observations,

such that ap, - Cp2 = 30 cm.The results for the instantaneous ambiguity resolu-

tion were as follows. For the 14 July data, for all epochs,the correct integer ambiguities could be obtained (suc-cess rate: 100%), as may normally be expected for a shortbaseline of 4-km length. However, for the 15 July data set,

for 6% of the epochs the ambiguities were not estimatedcorrectly (success rate: 94%), which is most likely caused

by larger, and for 6% of the epochs even non-negligible,ionospheric delays. So, from an ambiguity resolutionpoint of view, the traditional "short-baseline model," inwhich the ionospheric delays are neglected, seems to fail.

An easy way to account for these ionospheric delays is toextend model (1) with unknown parameters for theionospheric delays. However, due to a large number of

unknowns the probability of successful ambiguity resolu-tion becomes too poor.

Coordinate estimationIn Figure 3 the estimated coordinate components aregiven for the 5-h time span with the integer ambiguities

estimated with model (1) held fixed. The coordinates werealso instantaneously estimated, so for each epoch a new

set was obtained, except for the 6% of the epochs for whichthe integer ambiguities were wrongly estimated (for theseepochs no coordinate values are plotted in the figure).

The used ambiguity-fixed mathematical model perepoch is:

(2)

Figure 3 shows that for both days the coordinate timeseries are not quite constant as may be expected for a sta-tic baseline. This effect is even more pronounced for the

North-component on 15 July, especially from epoch 300.This non-constant behavior is most likely to be caused byrelative ionospheric delays, which seem to be insignifi-cant for the purpose of ambiguity resolution (as for

almost all epochs the correct ambiguities could be esti-mated), though it does appear to bias the ambiguity-fixedcoordinate solution. The biases on 15 July seem to be

larger than on the day before, presumably because of thesevere geomagnetic storm conditions.

For precise applications these ionosphere-biased

coordinate estimates are unacceptable, and again thenormal "short-baseline model" seems to fail. One way to

FIGURE 3. Instantaneous ambiguity-fined coordinate! for DYHi (reference BBIS at 4 km) estimated with themodel in which ionospheric delays are neglected, lime spans were 19:00-24:00 OPS on 14 July Cleft) and thesame time span on 15 July (right). Coordinate estimates are North, East, Up corrections with respect to the apriori values and +10 cm shifted for the North-coordinate and -10 cm shifted for the Up-coordinate to improvethe visibility of the graphs. For each component an empirical standard deviation is computed over all epochs andgives! in meters above the time series.

account for the ionospheric delays after ambiguity fixingis by extending the mathematical adjustment model withunknown parameters for the ionosphere:

(3)

Note that the stochastic part is the same as the one ofmodel (1).

Note, as per model (3), that use is made of the fre-quency dependence of the ionospheric delays. The coor-dinate solution of model (3) corresponds to a solution inwhich, after ambiguity resolution, the ionosphere-free lin-

ear combination of dual-frequency observations is taken.In Figure 4 the coordinates estimated with model (3)

are given. Comparing Figure 4 with Figure 3 reveals thatthe systematic behavior in the coordinates due to

unmodeled ionospheric delays has disappeared, exceptfor the Up-component. The long-term fluctuation visible

in the Up-component could be caused by unmodeledtropospheric delays as for the troposphere no unknownswere estimated in the processing (the data were only apriori corrected using the Saastamoinen model). Howev-

er, the horizontal coordinates are reasonably stable intime and the fluctuation remaining largely stems from

propagation of the measurement noise into the coordi-nate estimates. This propagation with model (3) is larger

than with model (2): Due to the presence of the ionos-

pheric unknowns the ambiguity-fixed coordinate preci-sion is about a factor of 4 worse than the precisionobtained when the ionospheric delays may be assumedabsent (Teunissen, 1998).

Ionospheric Corrections Generated from aPermanent GPS NetworkIn the previous section it was shown that under geomag-

netic storm conditions ionospheric delays might impactinstantaneous ambiguity resolution and coordinate esti-mation, for a 4-km baseline. A way to improve this pro-

cessing is the use of a permanent GPS network, if presentin the surveying area. In many countries such permanent

networks have been set up (e.g., Rizos, Han, & Chen,2000). Such a network can be used to generate and sup-

ply ionospheric corrections to users operating within thecoverage area.

In Figure 5 a method to generate these ionosphericcorrections is schematically given. The GPS data collect-ed by the permanent stations are processed according toa network adjustment, with the coordinates of the sta-tions held fixed to their precisely known values. From this

adjustment precise ionospheric delays between the per-manent stations can be obtained (in double-differencemode and for each epoch), as well as other information.

Next, these estimated ionospheric delays are used to spa-

tially predict the ionospheric delays at specified user sitesby performing a linear interpolation at the approximated

FIGURE 4. Instantaneous ambiguity-fined coordinates for DYHS (reference BGIS at 4 km) estimated with themodel in which unknowns for the ionospheric delays ere parameterized, lime spans were 19:00-24:00 SPS on14 July Cleft) and the time span on 13 July (right).

Instantaneous Precise GPi Positioning 33

FIGURE i. Generation of the ionosphericcorrections to he applied to the data of a useroperating within a permanent OPS network.

FIGURE S. Selected uses9 baselines (the "light"lines) within the permanent network consisting of

stations LSNJ-TRAK-WIDC-SS03-SNI1 (connectedHy the "dark" lines). Permanent station LINJ fsassigned as "master" reference station from whichthe double-differences in the network are related.Note that the dotted Sines are dependent baselines ofthe permanent network.

user location (see e.g., Odijk, van der Marel, & Song,

2000).

So with a permanent GPS network available users can

correct their GPS measurements for the ionospheric delay.

Instead of model (1) they should apply the following model:

(4)

with for I the interpolated correction from the perma-

nent network. Note that the stochastic part of model (1)

remains the same.

As mentioned before, the stations BGIS and DYHS,

which form the short baseline described in the previous

section, are in reality two permanent stations of the

SCIGN network. From this network another five stations

were selected, LINJ-WIDC-TRAK-SIO3-SNI1, to serve as a

network of permanent stations from which ionospheric

delays for the short baseline were generated. In Figure 6

the configuration of the stations is illustrated. The dis-

tances between the five permanent stations range from

about 100 to 200 km. From the estimated ionospheric

delays between the permanent stations the ionospheric

delays for user baseline BGIS-DYHS were interpolated

and these interpolated delays were used to correct the

phase and code data according to model (4).

For the purpose of instantaneous ambiguity resolu-

tion, the application of these corrections turned out to be

beneficial for the 15 July data: The ambiguity success rate

increased from 94% obtained with the model without cor-

rections (see previous section) to 100% with the ionos-

phere-corrected model (4). The success rate on 14 July

remained at 100%, with or without corrections applied.

Hedium-laseline Performance with IonosphericCorrections AppliedWith the short-baseline success in mind in a next step

interpolated ionospheric corrections were also generated

for a longer user baseline within the area of the perma-

nent network. It is known that for this type of "medium-

length" baselines the ionospheric delays are usually the

largest errors seriously hampering fast GPS positioning,

and therefore instantaneous ambiguity resolution with-

out a priori corrections is virtually impossible.

In this section the performance of instantaneous

ambiguity resolution and coordinate estimation will be

investigated for a 54-km baseline from (permanent) station

LINJ to assigned user station BRAN (see Figure 6) when

34 Odijk

FIGURE 7. Number of satellites for mediumbaseline LSNJ-BRAN on 14 July 2000 from18:00-24:00 UTC.

interpolated ionospheric corrections are applied. The samedays and time spans were used as for the short baseline.

In Figure 7 for baseline LINJ-BRAN the number oftracked satellites are given during the 5-h time span on 14

July. This number is at all epochs less or equal to thenumber of satellites tracked by the short baseline BGIS-DYHS (see Figure 2), as a consequence of the longer base-

line length. However, at all epochs the number of satel-lites is larger than four.

have been corrected by the interpolated network delays. Inorder to make this visualization, the ionospheric delaysover the 1-h time span were not based on the instanta-

neous ambiguity resolution results, as there were manyepochs with wrong ambiguities, but one ambiguity set wasestimated over the complete hour using their time-con-stancy in a batch processing. As a result, the plotted ionos-

pheric delays over all epochs are based on correct fixedambiguities and have therefore a high (few mm) precision.

In Figure 9 (a) the impact of the severe geomagneticstorm on 15 July is clearly visible: In general the magni-

tude of the real DD ionospheric delays in the data on 15July are larger then those on 14 July (note that on bothdays the same reference satellite was selected). Becausethe satellite-receiver geometry was approximately the

same for both days (as a consequence of the daily repeatof the satellite constellation), the difference in the esti-

mated DD ionospheric delays between the two daystherefore stems largely from a difference in ionosphericconditions. Note that the receiver-satellite geometry forthe 1-h time span is reflected in the skyplot in Figure 8.

Moreover, note that the ionospheric delays on 14 Julyare relatively smooth from one epoch to the next, but on15 July one can see scattering effects for PRN 23. This

scattering could be due to ionospheric scintillationwhich may occur occasionally during geomagneticstorms (Kunches, 2000).

To test the performance of epoch-by-epoch ambiguity

resolution, for both days ionospheric corrections weregenerated from the permanent network and applied tothe data of baseline LINJ-BRAN according to model (4).

The resulting ambiguity success rates were, however,disappointing. Despite the data being corrected, for 14

July the success rate was 66%, while for 15 July, the geo-magnetic storm day, it dropped to a low 36%. A further

investigation revealed that these poor success rates werecaused by significant residual ionospheric delays in the

data. Obviously the differences between the interpolatedcorrections and the real ionospheric delays in the datawere, for many epochs, too large to permit an estimation

of the correct integer ambiguities.To illustrate this, for 1 h of the 5-h time span,

21:00-22:00 UTC on both days, in Figure 9 double-differ-ence (DD) ionospheric delays are plotted: in Figure 9 (a)

the "real" DD ionospheric delays present in the data, andin Figure 9(b) the residual delays after the observations

08 L

FIGURE 8. Skyplot for station on 14 July for thetime span 21:00-22:00 UTC (cut-off elevation 10°).

Instantaneous Precise GPS Positioning 35

FIGURE 8(a). Rea! OD Ionospheric delays in the 11-observations on 14 July Cfeft) and 15 JuSy (right) for thetime span 21:00-22:00 Oie.

FIGURE 9b. Residual Creel minus interpolated) DO ionospheric delays in the LI-observations on 14 July (left)end 13 July (right) for the time span 21:00-22:00 UTO.

From Figure 9(b) it appears that on the geomagnetic

storm day, 15 July, the linearly interpolated ionosphericcorrections differ more from the real ionospheric delaysin the baseline data. For the 14 July time span the differ-

ences are smaller than 5 cm, while for the 15 July timespan there are differences as large as 15 cm. Due to such

large discrepancies instantaneous ambiguity resolutionwith model (4) becomes difficult.

Coorainati sstimsiianFor the epochs for which the integer ambiguities were

correctly resolved, in a next step the baseline coordinateswere estimated using model (4) but with the ambiguitiesheld fixed. In Figure 10 the coordinate time series over 5

h on both days are given.

Inspecting Figure 10, particularly on 15 July one can

hardly speak of a coordinate time series because for somany epochs no coordinates were estimated as a conse-quence of incorrect ambiguity sets. For the epochs with

correctly estimated integers, the estimated coordinatesalso on 14 July seem to be very biased due to significant

residual ionospheric delays.

Medium-BeseSine Performance with Stochasticionospheric Corrections AppliedFrom the results described in the previous section, it can

be concluded that under disturbed ionospheric condi-tions the ionospheric corrections as generated from the

permanent network are not precise enough to guaranteea high single-epoch ambiguity success rate, and respec-

tive coordinates which are not significantly biased byresidual ionospheric delays. In model (4) the ionosphericcorrections were considered to fully correct for the ionos-

pheric delays; however, the network interpolated correc-tions are not able to completely correct for it, as a resultof the ionospheric conditions and the distances betweenthe permanent network stations.

To account for these inaccurate ionospheric correc-tions, one could consider them to be stochastic instead of

deterministic. If the ionospheric corrections are assumedto have an a priori variance, <r? (undifferenced), thenpropagating this into the variance-covariance matrix ofmodel (4) results in:

Note that it can be proven that the baseline and ambiguity solution of model (5) may also be obtained using the fol-lowing model:

In contrast to model (5), model (6) solves explicitlyfor the ionospheric delays.

To successfully apply this technique of stochasticionosphere modeling, a proper choice of the ionospheric

standard deviation is important. Odijk (2000) suggested aprocedure to empirically assess this standard deviation

from the network of permanent stations. Hence itdepends on the baseline length and the actual ionos-pheric activity. Applying this procedure to our current

data sets, the following ionospheric standard deviationswere assessed for the medium-length baseline LINJ-

FIGURE 10. Instantaneous ambiguity-fixed coordinates for BRAN (reference U!\U at 04 km) estimated with themodel In which the observations are determsnistically corrected for the ionospheric delays. Ime spans were18:00-14:00 OPi on 14 July Cleft) and the same time span on 15 July (right).

Instantaneous Precise SPS Positioning 37

BRAN: for 14 July tr/ = 2 cm and for 15 July a standard

deviation twice as large: ai - 4 cm (note that these stan-dard deviations are for data in the undifferenced mode).

The performance of instantaneous ambiguity resolutionwith the stochastic ionospheric corrections improved sig-nificantly compared to the results without the stochasticmodeling. For 14 July the ambiguity success rate increased

from 66% to 95%, and for 15 July from 36% to 77%!Although there is a significant improvement in the

success rate, especially on 15 July the success rate is stillnot high enough if one aims at a success rate close to

100%. This stems from the fact that the proposed model(5) becomes weaker as the ionospheric standard devia-tion increases. In the limiting case, i.e., 07 = °°, the ionos-

pheric corrections do not contribute anymore to the solu-tion of the model and the results equal those of the modelin which the ionospheric delays are completely unknownparameters. So the ambiguity success rate is a decreasing

function of or. Considering the ionospheric activity on 15July and the length of this baseline LINJ-BRAN, the ionos-pheric standard deviation was assessed at 4 cm, resulting

in a single-epoch success-rate much lower than 100%.

Coordinate estimationAfter ambiguity resolution, for those epochs with correct-ly resolved integers final baseline coordinates were esti-

mated. For these solutions the same ionospheric stan-dard deviations were used as for the ambiguity resolution

step. In Figure 11 the instantaneous coordinate estimatesare plotted for the 5-h time spans on both days.

Comparing the coordinates in Figure 11 with those

of Figure 10, in which the ionospheric corrections weretreated as deterministic quantities, in Figure 11 a muchmore time-constant behavior is evident. However, in theUp-component there are some large jumps visible, espe-

cially around epochs 400 and 520 on both days. Also inthe North- and East-components jumps are visible,although much smaller than for the Up-component.These jumps can be explained when the satellite-geome-

try is considered. In Figure 7 it can be seen that the num-ber of satellites changes several times during the timespan, also at epochs 400 and 520, due to risings and set-

tings of the GPS satellites. As a consequence of thischanging geometry, the precision of the baseline coordi-nates changes. In Figure 12 the (ambiguity-fixed) stan-

dard deviations of the North-, East-, and Up-componentof BRAN are plotted. It can be seen that the precisionchanges due to the changing number of satellites, with

the vertical component the most sensitive comparedwith the horizontal components. Especially aroundepoch 520 the largest jump occurs in the Up-component.The number of satellites is reduced from 6 to 5 and it

seems that the satellite that sets at this epoch has a sig-nificant impact on the estimation of the Up-component.

Concerning the precision of the baseline coordinates

on average, with an ionospheric standard deviation of2-4 cm it is at cm-level: about 1-2 cm for the North- and

East-components and 2-5 cm for the Up-component.

FIBURE 11. instantaneous ambiguity-fixed coordinates for BRffl (reference UI\!J at 54 km) estimated with themodel in which the observations are stochastically corrected for the ionospheric delays. Ime spans were18:00-24:00 OPS on 14 July Cleft) and the same time span on 15 July (right).

FEGURE12. Formal precision of the instantaneousHaseiine components Worth, East, and Up for BRMlduring the time span 18:00-24:00 OTC on 14 July2000.

Short-Baseline Performance with Stochasticionospheric Corrections AppliedIn the preceding it was mentioned that the ambiguitysuccess rate for the short 4-km baseline BGIS-DYHS was94% on 15 July, without applying any ionospheric correc-

tions (on 14 July it was 100%). With deterministic correc-tions applied, the success rate increased to 100% for bothdays; so from this point of view stochastic corrections, as

applied to the medium-length baseline case, do not makeany significant contribution. However, if the ambiguity-

fixed coordinates are considered, there might be animprovement, as with stochastic corrections the influ-ence of residual ionospheric delays (due to differences ininterpolated and real ionospheric delays) might cancel,as was the case with the medium-length baseline. There-

fore ionospheric corrections were applied to this shortbaseline with the following ionospheric standard devia-tions: for 14 July, aj-2 mm; and for 15 July, ai= 3 mm.

Application of the stochastic corrections resulted in a

100% success rate for both days, the same as in the caseof the deterministic corrections.

Saoi'tlmate sstimationWith the resolved integer ambiguities held fixed, theresulting baseline coordinates are plotted in figure 13.

Comparing Figure 13 with Figure 3 it appears thatthe estimated coordinates in Figure 13 are not biased dueto residual ionospheric delays, as the coordinate esti-mates seem to be more constant in time. The solution

using model (3), in which the ionospheric delays areassumed to be completely unknown parameters, alsoresulted in ionospheric-free coordinates, though at alower precision level (see Figure 4). The precision of the

coordinates estimated with the stochastic corrections isbetter than those in Figure 4 by about a factor of 2.

The conclusion seems to be that for very short base-lines ionosphere weighting also improves the instanta-

FIGURE 13. Instantaneous amhiguity-fiMed coordinates for DYHS (reference BOIS at 4 km) estimated with themodel in which the observations are stochastically corrected for the Ionospheric delap. Hme spans were18:00-14:00 IPS on 14 July Cleft) and the same time span on 15 July (right).

Instantaneous Precise OPS Positioning 39

neous ambiguity resolution, and the resultant coordinatesolution, during periods of geomagnetic storms.

Horizontal Position ScattersFinally, in this section the previous showed coordinate

time series are presented in the form of horizontal posi-tion scatter plots (Figures 14 and 15). Figure 14 shows theestimated short-baseline coordinates and Figure 15 the

medium-baseline coordinates.Clearly visible in Figure 13 is that on both days the

positions estimated using the stochastic corrections aremuch more clustered and centered around zero than the

positions without these corrections. This effect is morepronounced for the medium-length baseline solutions,reflected in Figure 14, though in this figure the dark scat-ter reflect the positions estimated with the model usingcorrections that are deterministic.

Also these horizontal-position scatter plots illustratethe significant improvement in instantaneous relativeGPS positioning when stochastic ionospheric corrections

are used.

In this article the effect of a geomagnetic storm on high-precision GPS relative positioning was discussed. Such astorm increases the magnitude and variability of the

ionospheric delays in the GPS phase and code data con-siderably, not only for baselines of medium length (about50 km), but also for short baselines (< 10 km). Data from

two geomagnetic storm days were analyzed: 14 July 2000with minor storm levels, and 15 July 2000 with severe

storm activity. For a 4-km baseline it was shown that forthe purpose of fast (instantaneous: using one epoch ofdata) integer ambiguity resolution and precise coordi-

nate estimation, the relative ionospheric delays may notbe neglected during severe geomagnetic storm periods,in contrast to what is usually done in GPS practice.

The availability of a permanent GPS network in the

surveying area may improve the results of a user's preciseGPS positioning. From the continuous data of such a net-work ionospheric corrections at the approximated user'sposition may be generated and provided to the user.

Results show that the user's instantaneous positioningimproves; however, for the medium-length baseline theresults are still not sufficiently improved. A further

improvement is gained when the ionospheric correctionsfrom the permanent network are processed as stochastic,instead of deterministic, quantities, resulting in a 95%

instantaneous ambiguity resolution success rate for themedium-length baseline during minor storm periods.However, the success rate dropped to 77% during thesevere storm period. During the severe storm, it seems

FIGURE 14. Horizontal coordinate corrections [m] for receiver DYHS (reference B01S) for 14 July (left) and15 July (right). The dark dots represent the instantaneous solutions ohtained from the model with integerambiguities held ffMed and no ionospheric corrections were used. The light dots are the solutions obtained withthe model with stochastic ionosphere corrections.

40 Odijk

FIQURE 15. Horizontal coordinate corrections (m) for receiver BRAN (reference LINJ) for 14 July (left) and15 July (right). The dark dots represent the instantaneous solutions obtained from the model with integerambiguities held fixed and deterministic ionospheric corrections were used. The Sight dots ere the solutionsobtained with the model with stochastic ionosphere corrections.

that the variability of the ionospheric delays is too largecompared to the station separation of the permanentnetwork.

Of course, if instead of a time span of one singleepoch the time span is lengthened and use is made of thetime-constancy of the ambiguities, residual ionospheric

effects average out to a certain extent, and the success ofambiguity resolution may increase, especially for themedium-length baseline. However, in this article the

attention was focused on instantaneous data processing.

Finally, one has to bear in mind that the averageoccurrence of severe geomagnetic storms is only 4 timesduring a solar cycle, though minor and major geomag-

netic storms occur much more frequently. GPS usershave to be aware that this will be the case even a few years

after the maximum of the current solar cycle.

ACHSVOWLEDGMTIThis study was supported by the Australian ResearchCouncil (ARC). The Southern California Integrated GPS

Network, and its sponsors the W.M. Keck Foundation,

NASA, NSF, USGS, and SCEC, are acknowledged for pro-viding the data used in this study. Professor Chris Rizos ofthe University of New South Wales is thanked for critical-

ly reading the manuscript.

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Odijk, D. (2000). Weighting ionospheric corrections to improvefast GPS positioning over medium distances. In Proc. ION GPS-2000 (pp. 1113-1124). Salt Lake City, Utah, USA, September 19-22.

Odijk, D., van der Marel, H., & Song, I. (2000). Precise GPS posi-tioning by applying ionospheric corrections from an Active Con-trol Network. GPS Solutions, 3(3), 49-57.

Rizos, C., Han, S., & Chen, H.Y. (2000). Regional-scale multiplereference stations for carrier phase-based GPS positioning: Acorrection generation algorithm. Earth Planets Space, 52(10),795-800.

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BIOGRAPHYDennis Odijk graduated from the Department of Mathe-

matical Geodesy and Positioning of the Delft Universityof Technology in The Netherlands. He is currently a Ph.D.

Instantaneous Precise OPS Positioning 41

student at the same department, where he is engaged in Navigation and Positioning Group of the University ofthe development of GPS data processing strategies for New South Wales in Sydney, Australia,

medium scaled networks with an emphasis on ambiguity Peer Review: This paper has been peer-reviewed andfixing and modeling of the ionospheric delays. At the accepted for publication according to the guidelines pro-time of writing this paper, he was staving at the Satellite vided in the Information for Contributors.

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